It is a job interview question.
So, what's the value of a vanilla European call option of infinite maturity, and a given strike, vol, interest rate, spot price.
I think, the answer should be "zero".
The contract never pays, because infinite maturity will never be reached.
It should not be equal to the spot price, which BS formula suggests in the limit T goes to infinity, I think.
If it were an American call with infinite maturity, the price could be anywhere between S and S-K.